The Complexity of Dynamic Least-Squares Regression

We settle the complexity of dynamic least-squares regression (LSR), where rows and labels (A(t), b(t)) can be adaptively inserted and/or deleted, and the goal is to efficiently maintain an ε-approximate solution to min_x(t) | A(t) x(t) - b(t) |2 for all t∈ [T]. We prove sharp separations (d2-o(1) vs. ~ d) between the amortized update time of: (i) Fully vs. Partially dynamic 0.01-LSR; (ii) High vs. low-accuracy LSR in the partially-dynamic (insertion-only) setting. Our lower bounds follow from a gap-amplification reduction -- reminiscent of iterative refinement -- rom the exact version of the Online Matrix Vector Conjecture (OMv) [HKNS15], to constant approximate OMv over the reals, where the i-th online product Hv(i) only needs to be computed to 0.1-relative error. All previous fine-grained reductions from OMv to its approximate versions only show hardness for inverse polynomial approximation ε = n-ω(1) (additive or multiplicative) . This result is of independent interest in fine-grained complexity and for the investigation of the OMv Conjecture, which is still widely open.